Subgroups — #12009103

Order 1280.

330 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2960	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#672	C₂	1	1	yes	maximal, characteristic
3	C₄×#3944	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊#672	C₂	1	4	yes	maximal, complemented
5	C₅×#533	C₄	1	1	yes	characteristic
6	C₅×#446	—	2	1	no	—
7	C₅×#446	C₄	1	2	yes	—
8	C₅×C₈×Q₈	—	2	1	no	—
9	C₅×C₈×Q₈	C₂²	1	2	yes	—
10	C₅×#424	—	2	1	no	—
11	C₅×#423	C₄	1	1	yes	characteristic
12	C₅×#405	—	2	1	no	—
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₂×#3944	—	2	1	no	—
15	C₂²×#672	—	2	1	no	—
16	C₄⋊#672	—	2	1	no	—
17	C₄⋊#672	C₂²	1	2	yes	—
18	#64681	—	2	1	no	—
19	C₄×#672	C₂²	1	2	yes	—
20	#11606	—	5	1	no	maximal, Sylow
21	C₂×C₅×M₁₆	—	2	1	no	—
22	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
24	C₅×C₄⋊Q₈	C₈	1	1	yes	characteristic
25	C₅×C₄⋊Q₈	C₈	1	1	yes	characteristic
26	C₅×#125	—	2	1	no	—
27	C₅×C₂²⋊Q₈	—	2	1	no	—
28	C₅×C₂²⋊Q₈	C₈	1	2	yes	—
29	C₄×C₅×Q₈	—	2	1	no	—
30	C₄×C₅×Q₈	C₂×C₄	1	2	yes	—
31	C₅×#117	—	2	1	no	—
32	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
33	C₅×C₄⋊C₈	—	2	2	no	—
34	C₅×C₄⋊C₈	—	2	2	no	—
35	C₅×C₄⋊C₈	—	2	1	no	—
36	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
37	C₅×#97	—	2	2	no	—
38	C₄×C₅×C₈	—	2	2	no	—
39	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
40	C₄×C₅×C₈	D₄	1	1	yes	characteristic
41	C₄×C₅×C₈	D₄	1	1	yes	characteristic
42	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
43	#3944	—	2	4	no	—
44	C₂×#672	—	2	2	no	—
45	C₂×#672	—	2	2	no	—
46	C₂×#672	—	4	2	no	—
47	#2960	D₅	1	1	yes	characteristic
48	C₂×C₄⋊C₁₆	—	5	1	no	—
49	C₄×M₃₂	—	5	1	no	—
50	Q₈⋊C₁₆	—	5	4	no	—
51	C₂×C₅×Q₈	—	2	1	no	—
52	C₂×C₅×Q₈	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
56	C₅×M₁₆	—	4	2	no	—
57	C₂×C₅×C₈	—	2	2	no	—
58	C₂×C₅×C₈	—	2	2	no	—
59	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₅×C₈	SD₁₆	1	2	yes	—
62	C₂×C₅×C₈	Q₁₆	1	2	yes	—
63	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
64	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
65	C₅×C₄⋊C₄	—	2	1	no	—
66	C₅×C₄⋊C₄	—	2	1	no	—
67	C₅×C₄⋊C₄	—	2	1	no	—
68	C₅×C₄⋊C₄	—	2	2	no	—
69	C₅×C₄⋊C₄	—	2	2	no	—
70	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
71	C₅×C₂²⋊C₄	—	2	2	no	—
72	C₄²×C₅	—	2	2	no	—
73	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
74	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
75	C₄²×C₅	M₁₆	1	1	yes	characteristic
76	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
77	#672	—	2	4	no	—
78	#672	—	4	2	no	—
79	#533	Dic₅	1	1	yes	characteristic
80	C₂×M₃₂	—	10	1	no	—
81	C₂²×C₁₆	—	10	1	no	—
82	#446	—	2	1	no	—
83	#446	Dic₅	1	2	yes	—
84	C₈×Q₈	—	2	1	no	—
85	C₈×Q₈	C₂×D₅	1	2	yes	—
86	#424	—	2	1	no	—
87	#423	Dic₅	1	1	yes	characteristic
88	#405	—	2	1	no	—
89	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
90	C₄⋊C₁₆	—	5	2	no	—
91	C₄⋊C₁₆	—	10	1	no	—
92	#346	—	10	1	no	—
93	C₄×C₁₆	—	5	2	no	—
94	C₂³×C₅	C₂²⋊C₈	1	1	yes	characteristic
95	C₅×Q₈	—	2	4	no	—
96	C₅×Q₈	—	4	2	no	—
97	C₂×C₄×C₅	—	2	2	no	—
98	C₂×C₄×C₅	—	2	2	no	—
99	C₂×C₄×C₅	—	2	2	no	—
100	C₂×C₄×C₅	—	4	1	no	—
101	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
102	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
103	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
104	C₂×C₄×C₅	#103	1	2	yes	—
105	C₂×C₄×C₅	#104	1	2	yes	—
106	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
109	C₅×C₈	—	2	1	no	—
110	C₅×C₈	—	2	1	no	—
111	C₅×C₈	—	4	2	no	—
112	C₅×C₈	#103	1	2	yes	—
113	C₅×C₈	C₂×SD₁₆	1	1	yes	characteristic
114	C₅×C₈	C₂×Q₁₆	1	1	yes	characteristic
115	C₂×M₁₆	—	2	1	no	—
116	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
117	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
118	C₄⋊Q₈	#168	1	1	yes	characteristic
119	C₄⋊Q₈	#168	1	1	yes	characteristic
120	#125	—	2	1	no	—
121	C₂²⋊Q₈	—	2	1	no	—
122	C₂²⋊Q₈	#168	1	2	yes	—
123	C₄×Q₈	—	2	1	no	—
124	C₄×Q₈	C₂×Dic₅	1	2	yes	—
125	#117	—	2	1	no	—
126	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
127	M₃₂	—	10	4	no	—
128	C₂×C₁₆	—	10	2	no	—
129	C₂×C₁₆	—	10	2	no	—
130	C₂×C₁₆	—	20	2	no	—
131	C₄⋊C₈	—	2	2	no	—
132	C₄⋊C₈	—	2	2	no	—
133	C₄⋊C₈	—	2	1	no	—
134	C₄⋊C₈	C₂×Dic₅	1	2	yes	—
135	#97	—	2	2	no	—
136	C₄×C₈	—	2	2	no	—
137	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
138	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
139	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
140	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
141	C₂²×C₅	—	2	1	no	—
142	C₂²×C₅	—	2	1	no	—
143	C₂²×C₅	Q₈⋊C₈	1	2	yes	—
144	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
145	C₄×C₅	—	2	1	no	—
146	C₄×C₅	—	2	1	no	—
147	C₄×C₅	—	2	1	no	—
148	C₄×C₅	—	4	2	no	—
149	C₄×C₅	—	4	2	no	—
150	C₄×C₅	Q₈⋊C₈	1	2	yes	—
151	C₄×C₅	C₂×#103	1	1	yes	characteristic
152	C₄×C₅	C₂×#104	1	1	yes	characteristic
153	C₄×C₅	#504	1	2	yes	—
154	C₂×Q₈	—	2	1	no	—
155	C₂×Q₈	C₂×#168	1	2	yes	—
156	C₂²×C₄	C₂×#168	1	1	yes	characteristic
157	C₂²×C₄	#681	1	1	yes	characteristic
158	C₂²×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
159	M₁₆	—	4	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	—	2	2	no	—
162	C₂×C₈	#688	1	2	yes	—
163	C₂×C₈	#689	1	2	yes	—
164	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
165	C₂×C₈	C₂²⋊Dic₅	1	1	yes	characteristic
166	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
167	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
168	C₄⋊C₄	—	2	1	no	—
169	C₄⋊C₄	—	2	1	no	—
170	C₄⋊C₄	—	2	1	no	—
171	C₄⋊C₄	—	2	2	no	—
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	C₂×#168	1	2	yes	—
174	C₂²⋊C₄	—	2	2	no	—
175	C₄²	—	2	2	no	—
176	C₄²	C₂×#168	1	1	yes	characteristic
177	C₄²	#681	1	1	yes	characteristic
178	C₄²	C₂²⋊Dic₅	1	1	yes	characteristic
179	C₄²	C₂²×Dic₅	1	1	yes	characteristic
180	C₁₆	—	10	4	no	—
181	C₁₆	—	20	2	no	—
182	C₂×C₅	—	2	2	no	—
183	C₂×C₅	C₂×Q₈⋊C₈	1	1	yes	characteristic
184	C₂×C₅	#2113	1	1	yes	characteristic
185	C₂×C₅	#2117	1	1	yes	characteristic
186	C₂³	C₂²⋊#168	1	1	yes	characteristic
187	Q₈	—	2	4	no	—
188	Q₈	—	4	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	4	1	no	—
193	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
194	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
195	C₂×C₄	C₂²⋊#168	1	1	yes	characteristic
196	C₂×C₄	#3967	1	2	yes	—
197	C₂×C₄	#3969	1	2	yes	—
198	C₂×C₄	C₂²×#168	1	1	yes	characteristic
199	C₂×C₄	C₂×#681	1	1	yes	characteristic
200	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
201	C₈	—	2	1	no	—
202	C₈	—	2	1	no	—
203	C₈	—	4	2	no	—
204	C₈	#3967	1	2	yes	—
205	C₈	C₂×#688	1	1	yes	characteristic
206	C₈	C₂×#689	1	1	yes	characteristic
207	C₅	#11606	1	1	yes	characteristic, complemented, Sylow
208	C₂²	—	2	1	no	—
209	C₂²	—	2	1	no	—
210	C₂²	Q₈⋊#168	1	2	yes	—
211	C₂²	C₂×C₂²⋊#168	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	4	2	no	—
216	C₄	—	4	2	no	—
217	C₄	Q₈⋊#168	1	2	yes	—
218	C₄	#65429	1	2	yes	—
219	C₄	C₂×#3967	1	1	yes	characteristic
220	C₄	C₂×#3969	1	1	yes	characteristic
221	C₂	—	2	2	no	—
222	C₂	C₂×Q₈⋊#168	1	1	yes	characteristic
223	C₂	#10602797	1	1	yes	characteristic
224	C₂	#10602799	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅
commutator	C₂×C₄×C₅	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅
Fitting	C₅×#2960	C₂
socle	C₂²×C₅	C₂×C₂²⋊C₈
prime radical	C₂³×C₅	C₂²⋊C₈
cosocle	C₄×C₅×C₈	C₂³
hypercenter	#2960	D₅
hypocenter	C₅	#11606
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2960

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. Q₈⋊#672	C₂	semidirect
207. C₅	#11606	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅×#2960	2
C₂×C₄⋊#672	2
C₄×#3944	2
Q₈⋊#672	2
#11606	5